ARTIFICIAL ELECTRIC LINES WITH MUTUAL INDUCTANCE. 197 



3. General Definitions. — 



2i = complex impedance of the series elements. 



Z2 = complex impedance of the shunt elements. 



M= mutual inductance between adjacent series elements, esti- 

 mated positive when the coils are wound in the same sense. 



99 = retardation angle of current per section of the line = the 

 angle of lag of steady-state current in any section behind 

 the current in the preceding section. 



T = time lag of current in seconds per section. 



a = real attenuation constant of current per section of the line 

 = the exponent in the factor e~°, by which factor the 

 steady-state current-amplitude in any series element of 

 an infinite or non-reflective line must be multiplied to 

 give the current-amplitude in the next series element. 



Zi = complex surge impedance of the line = complex impedance 

 of a line of an infinite number of sections or a line with 

 non-reflective output impedance. 



03 = angular velocity of impressed sinusoidal e.m.f. in radians 

 per second. 



4. Abbreviations and Subsidiary Notation. — The following 

 notation taken from Electric Oscillations and Electric Waves with slight 

 augmentation will be employed. 



p -f ju, = r"^^ (1) 



where P and U are real quantities; 



1 - L'2 _ pi 



r = ^ ;• (2) 



h = U/V; (3) 



f(h) = ^yj±^W±l, ,w = +V+Z^Ii. (4) 



Other notation will be introduced below in Section 10. 



5. Equations for a and (p for Steady-State Current. — It is 

 proved in Electric Oscillations and Electric JJ^aves, equations (49) and 

 (50), p. 296, that, for the steady-state current under the action of an 



